Oscar Zariski

Oscar Zariski (1899-1986) did work of fundamental importance in
the area of algebraic geometry. This subject, which deals with
the solution sets of polynomial equations, can be traced back
to Descartes but only developed into an important
mathematical discipline in the nineteenth and twentieth centuries.
By the early twentieth century, the "Italian school" of algebraic
geometers had accumulated many interesting results but had become
mired in foundational problems. Zariski, an adopted member
of the Italian school, reformulated the subject in in terms of
modern algebra and provided the basis for its twentieth-century
development.

Zariski was born Ascher Zaritsky (he Westernized his name while in
Italy) in the city of Kobrin, then part of the Russian Empire.
His father, a Taldmudic scholar, died when he was two, leaving
his mother Hannah to support seven children. She did this by running
a store, and in fact the family became one of the richest in Kobrin.
Zariski was able to attend a gymnasium (prep school), where his
mathematical ability became evident. During the First World War
he lived with his brother and attended gymnasium in Chernigov, while
Kobrin was under German occupation; he was unable to see his mother
again until 1918. In 1918-20 he attended the University of Kiev during
the Russian Civil War. Zariski attended classes in the midst
of street battles; once he was wounded when Ukranian soldiers fired on
students. After the Communists took Kiev and abolished examinations,
he secretly went to the homes of professors to take them.

In 1920 Zariski again went home to Kobrin. The peace treaty between
Russia and newly independent Poland put Kobrin in Poland, forcing
Zariski to choose between Russian and Polish citizenship. Although
he was an idealistic socialist, Zariski wanted to pursue mathematics
in Western Europe and chose Polish nationality so he could get a passport
quickly. He was attracted to Italy, which waived university tuition for
foreigners and had a low cost of living. He first tried the University
of Pisa, but after finding the mathematics department there inferior to
Kiev's he moved on to the University of Rome.

In Rome, Zariski blossomed as a mathematician under the influence of
three great members of the Italian school of algebraic geometry: Castelnuovo,
Enriques, and Severi. Castelnuovo gave him his thesis problem (on solution
of equations by radicals), and Zariski completed his doctorate in 1924.
In the same year he married Yole Cagli, an Italian literature student.
The couple lived in Rome, at first in a rented apartment and later with
Yole's parents; their first child was born in 1925. Zariski, as a
Jewish socialist with Communist sympathies, was unwilling to become a
citizen of Fascist Italy and so could not join the university faculty; he
juggled odd jobs while Yole taught full-time. Things became a little
easier in 1926 when he received a Rockefeller fellowship. After he
was turned down for a Russian visa, Zariski decided to seek his fortune
in America. Castelnuovo contacted Lefschetz,
who agreed to support Zariski's application for a fellowship at Johns
Hopkins University.

Lacking money and a visa for Yole and the baby, the Zariskis reluctantly
separated for a year while Oscar established himself in America. This
he was able to do, though he found the Hopkins mathematics department
rather dull. Fortunately the distinguished algebraic geometer
A. B. Coble was visiting for a year, and Zariski also communicated with
Lefschetz, who enthusiastically supported
his work. Zarkiski became interested in using methods of algebraic
topology introduced by Poincaré,
particularly the fundamental group.
By spring 1928 he had the offer of a permanent position at
Hopkins, and on the strength of this was able to bring his family to
America. Zariski was to remain at Hopkins until 1946, though he faced
an unsympathetic adminstration and heavy teaching load. In 1935 he
published Algebraic Surfaces, a mathematical classic which summarized
the work of the Italian School. Ironically, it was during the writing of
this book that Zariski became "disgusted" with the Italian methods and
their lack of rigor, and started on his project of rebuilding algebraic
geometry on the foundation of modern commutative algebra, particularly
the work of Noether and Krull.

Zariski's pathbreaking work earned him a Guggenheim fellowship for 1939-40.
His plans to use the Guggenheim to visit England in fall 1939 were disrupted
by the outbreak of the Second World War, which also cut off contact with
his family in Poland. Zariski drove across the country to Caltech in early
1940 and spent 1940-41 visiting at Harvard. After this, it was hard to return
to an 18-hour teaching load at Hopkins.
His importance as a mathematician was recognized, but there were few positions
available during the war. At the invitation of the State Department he
visited São Paulo in 1945, where he was able to work with his friend
and intellectual sparring partner André Weil. It was in Brazil that he
learned of the deaths of most of his Polish relatives, including his
mother, at the hands of the Nazis. When he returned to the U. S., he
resigned his position at Hopkins to take one at the University of Illinois;
a year later he accepted a permanent position at Harvard, where he would
spend the rest of his career.

Zariski made Harvard a world center for algebraic geometry in much the
same way that Lefschetz made Princeton a
world center for topology. Among his students were such important algebraic
geometers as Abhyankar, Hironaka, Mumford, Michael Artin, Hartshorne,
and Kleiman. Together with Pierre Samuel he wrote Commutative Algebra,
a two-volume work which is still a standard textbook.
In the late 1950's and 1960's, algebraic geometry underwent
another transformation with the introduction of new methods by
Grothendieck and Serre. Zariski later wrote, "what I myself had called
abstract turned out to be a very very concrete brand of mathematics."
To some extent this new revolution left Zariski behind, though he was
happy to see his students use the new methods to go beyond his work,
as when Hironaka established
resolution of singularities in arbitrary dimension (for a ground field
of characteristic zero).